Historically, the Weierstrass function is important because it was the first published example (1872) to challenge the notion that every continuous function was differentiable except on a set of isolated points.[1]

The minimum value of , which satisfies these constraints is . This construction, along with the proof that the function is nowhere differentiable, was first given by Weierstrass in a paper presented to the Königliche Akademie der Wissenschaften on 18 July 1872.[2][3][4]

The proof that this function is continuous everywhere is not difficult. Since the terms of the infinite series which defines it are bounded by ±an and this has finite sum for 0 < a < 1, convergence of the sum of the terms is uniform by the Weierstrass M-test with Mn = an. Since each partial sum is continuous and the uniform limit of continuous functions is continuous, it follows f is continuous.

To prove that f is nowhere differentiable, we consider a point and show that the function is not differentiable at that point. To do this, we construct two sequences of points xn and x′n which both converge to x, having the property that

where "lim sup", and "lim inf" denote limit superior and limit inferior, respectively, of the sequence. Naïvely it might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be "small" in some sense. According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that this was true. This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiable points is something other than a countable set of points. Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue null set (Rademacher's theorem). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz and has other nice properties.

The Weierstrass function could perhaps be described as one of the very first fractals studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone. The Hausdorff dimension of the graph of the classical Weierstrass function is bounded above by 2 + ln(a)/ln(b), (where a and b are the constants in the construction above) and is generally believed to be exactly that value, but this had not been proven rigorously.[5][6] Notice that 1 < D < 2 if ab > 1.

The term Weierstrass function is often used in real analysis to refer to any function with similar properties and construction to Weierstrass' original example. For example, the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. G. H. Hardy showed that the function of the above construction is nowhere differentiable with the assumptions 0 < a < 1, ab ≥ 1.[7]

In a measure-theoretic sense: when the space C([0, 1]; R) is equipped with classical Wiener measureγ, the collection of functions that are differentiable at even a single point of [0, 1] has γ-measure zero. The same is true even if one takes finite-dimensional "slices" of C([0, 1]; R): the nowhere-differentiable functions form a prevalent subset of C([0, 1]; R).

^At least two researchers formulated continuous, nowhere differentiable functions before Weierstrass, but their findings were not published in their lifetimes. Around 1831, Bernard Bolzano (1781 - 1848), a Czech mathematician, philosopher, and Catholic priest, constructed such a function; however, it was not published until 1922. See:

^On page 560 of the 1872 Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (Monthly Reports of the Royal Prussian Academy of Science in Berlin), there is a brief mention that on July 18th, "Hr. Weierstrass las über stetige Funktionen ohne bestimmte Differentialquotienten" (Mr. Weierstrass read [a paper] about continuous functions without definite [i.e., well-defined] derivatives [to members of the Academy]). However, Weierstrass's paper was not published in the Monatsberichte.